Born in Whittier, California, USA, Hancock first came to the United Kingdom at the end of the 1988 speedway season to ride exhibition races with fellow 18 year old Californian Billy Hamill. It was during this time that Hancock agreed a deal to ride for Cradley Heath the following year – the same team that his mentor Bruce Penhall used to ride for. Hancock was an instant success for Cradley in the British League. He also won gold medals for the USA in the 1992 World Pairs and World Team Cup. However due to injury and problems with the American Motorcyclist Association, Hancock was not able to contest in the Individual World Championship until 1993. That year he went through to the World Final in Germany where he finished last place. In 1994, Hancock again qualified for the last ever 'one off' World Final. He went into his last race needing a victory to become world champion but he finished third in the race and fourth overall on the night.

In 1995 Hancock finished in 4th place in the first year of the Grand Prix (GP). He won the final round, the British GP at the Hackney Wick Stadium. The following year he joined with fellow American and Cradley teammate Billy Hamill to form Team Exide. With this new found sponsorship, the two young Californian's began to dominate world speedway. In 1996 Hamill won the world title and Hancock finished with the bronze medal in third place. In 1997 Hancock moved from Cradley Heath (due to the club's closure) to the Coventry Bees. He won the first GP of the year in Prague and continued the season in the same form. He won the world title (with Hamill finishing second for a Team Exide one two) and won the Elite League Riders' Championship.

In 1998 to Grand Prix format changed to a more cut throat elimination system. Hancock did not fare so well and finished the year in sixth place but won the World Team Cup with Hamill and Sam Ermolenko. The next year was even worse for Hancock and he finished in 11th place in the Grand Prix. 2000 saw Hancock win his first Grand Prix since his title winning season and he finished in fifth place and in 2001 he finished in 13th place. In 2002 Hancock won the last Grand Prix round in Australia and finished 6th overall. He went one better in 2003 finishing in 5th place after again winning the final round, this time in Haamar, Norway. The following season Hancock was back amongst the medals, finishing the season in third place. That year he won the British GP at the Millennium Stadium in Cardiff. In 2005 Hancock slipped back to fifth in the world unable to win a GP. A 36-year old Hancock finished the Grand Prix series in second place in 2006 and remained unbeaten in that year's World Cup but the USA did not qualify for the final as the rest of the team struggled. In 2007 Hancock finished in sixth place. Although he failed to win a GP, he finished in second place on three occasions. In 2008 Hancock finished fourth overall in the Grand Prix series. He was on the podium four times, winning the Polish Grand Prix in Bydgoszcz.

Greg finished 1st place in the 2011 Czech, British and Nordic Grand Prix. He secured the 2011 World Championship during the Croatian Grand Prix on 24 September, reclaiming the title after a record 14 years at the age of 41 years and 113 days. Hancock clinched the 2014 World Championship during the Polish Grand Prix on October 11, 2014 and broke his own record of oldest title winner at 44 years and 130 days.